Existence of the universal attractor for the 3-D viscous primitive equations of large-scale moist atmosphere

Abstract

In this paper, we consider the initial-boundary value problem for the three-dimensional viscous primitive equations of largescale moist atmosphere which are used to describe the turbulent behavior of long-term weather prediction and climate changes. By obtaining the existence and uniqueness of global strong solutions for the problem and studying the long-time behavior of strong solutions, we prove the existence of the universal attractor for the dynamical system generated by the primitive equations of largescale moist atmosphere.

On the Stability of Weak Solution for Compressible Primitive Equations

“…Proof From the above estimate (12), the bound of √ ξ n is obviously obtained. Using the continuity equation, we write ∂ t ξ n = − 1 2 ξ n ∂ x u n − u n ∂ x ξ n − ξ n ∂ z w n = 1 2 ξ n ∂ x u n − ∂ x (u n ξ n ) − ξ n ∂ z w n .…”

“…In [23], Temam and Ziane considered the local existence of strong solutions for the PEs of the atmosphere, the ocean and the coupled atmosphere-ocean. In [10][11][12], Guo and Huang proved the existence and uniqueness of global strong solutions for the three dimensional viscous PEs. Bresch et al [1] got the existence and uniqueness of weak solutions for the two-dimensional hydrostatic Navier-Stokes equations.…”

Global well-posedness for passively transported nonlinear moisture dynamics with phase changes

“…Plugging (5.14) and (5.15) into (5.13), and noticing that η = 0 onΓ ℓ \ ∂O, we deduce 16) for any φ ∈ H 1 (Ω). Define the truncation U =ũη, and denote 19) where N is the unit outward normal vector at the boundary ∂O.…”

“…The rigorous analysis was carried out in [25] for both the finite-time relaxation system and the instantaneous relaxation limiting system, including the global well-posedness and the strong convergence of the relaxation limit. Some moisture models without phase changes coupled to the primitive equations were also considered by Guo and Huang [16,17].…”

Formulation of the equations of the humid atmosphere in the context of variational inequalities

“…The mathematical theory of the equations of the humid atmosphere [16,28] has been initiated in [26,27] and more recently in [20,21]. However, in these references, saturation of water vapor in the air is not accounted for, so that the equation for the concentration q of water vapor in the air is a mere transport equation.…”